Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 431In terms of complex coordinates we can write the basis (3.239) asT p M C : {∂ z 1| p , ..., ∂ z d| p , ∂¯z 1| p , ..., ∂¯z d| p }.From the point of view of real vector spaces, ∂ x j | p and i∂ x j | p would beconsidered linearly independent and hence T p M C has real dimension 4D.In exact analogy with the real case, we can define the dual to T p M C ,which we denote by T ∗ p M C = T ∗ p M ⊗ C, with the one–forms basisT ∗ p M C : {dz 1 | p , ..., dz d | p , d¯z 1 | p , ..., d¯z d | p }.For certain types of complex manifolds M, it is worthwhile to refine thedefinition of the complexified tangent and cotangent spaces, which pullsapart the holomorphic and anti–holomorphic directions in each of thesetwo vector spaces. That is, we can writeT p M C = T p M (1,0) ⊕ T p M (0,1) ,where T p M (1,0) is the vector space spanned by {∂ z 1| p , ..., ∂ z d| p } andT p M (0,1) is the vector space spanned by {∂¯z 1| p , ..., ∂¯z d| p }. Similarly, wecan writeT ∗ p M C = T ∗ p M (1,0) ⊕ T ∗ p M (0,1) ,where Tp ∗ M (1,0) is the vector space spanned by {dz 1 | p , ..., dz d | p } andTp ∗ M (0,1) is the vector space spanned by {d¯z 1 | p , ..., d¯z d | p }. We call T p M (1,0)the holomorphic tangent space; it has complex dimension d and we callTp ∗ M 1,0 the holomorphic cotangent space. It also has complex dimensiond. Their complements are known as the anti–holomorphic tangent andcotangent spaces respectively [Greene (1996)].Now, a complex vector bundle is a vector bundle π : E → M whose fiberbundle π −1 (x) is a complex vector space. It is not necessarily a complexmanifold, even if its base manifold M is a complex manifold. If a complexvector bundle also has the structure of a complex manifold, and isholomorphic, then it is called a holomorphic vector bundle.3.14.1 Complex Metrics: Hermitian and KählerIf M is a complex manifold, there is a natural extension of the metric g toa mapg : T p M C × T p M C → C,